Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate...
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![Page 1: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/1.jpg)
Proving (or disproving) the Null Hypothesis
1. From the sample and hypothesis benchmark we calculate a test statistic
2. From the hypothesis and degree of confidence we calculate a critical region, bounded by critical values
3. If the test statistic lies within the critical region, we reject the null hypothesis
1. Otherwise we fail to reject it.
![Page 2: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/2.jpg)
1. Test Statistic
• Naturally, we have to sample, then calculateand s.
• From these two values, the sample size, and our hypothesis value, we calculate the test statistic:
• Where μ is our hypothetical benchmark
X
Xz
s
n
![Page 3: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/3.jpg)
For example
• People who eat five servings of fruit and veggies daily have an average life span of 80 years. Our sample of 120 people who has a mean lifespan is 82.5 years (standard deviation is 4.6 years)
• South students have an IQ greater than 115. Our sample of 36 students, their mean IQ is 108 years (standard deviation is 18)
![Page 4: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/4.jpg)
Your turn
• American cars have an average repair cost greater than $600– The mean of a sample of 45 cars was $520 with a standard
deviation of $120.
• The average New Jersey property tax bill is $6,500. – The mean of a sample of 65 homes was $6,700 with a
standard deviation of $3,200.
![Page 5: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/5.jpg)
Homework: Find the test statistic
1. H0 = 110, X-bar = 118, s = 12, n = 45
2. H0 ≥ 6.5, X-bar = 7, s = 1.2, n = 32
3. H0 = 45, X-bar = 50, s = 12, n = 100
4. H0 ≤ 1.1, X-bar = 0.8, s = 0.12, n = 60
5. H0 ≤ 12000, X-bar = 11800, s = 1002, n = 35
6. The average math SAT scores is at least 550– Sample mean is 535, standard deviation = 102, n = 90
7. The average GPA is 3.0– Sample mean is 3.05, standard deviation = 1.02, n = 30
![Page 6: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/6.jpg)
2. Critical Regions
• Area of the standard normal curve outside the degree of confidence– AKA Significance level
– α = 1 – degree of confidence
– Probability that we will reject H0 even though it’s true
• Bounded by the critical value(s)– Will be compared with the test statistic
– Found in the tables
• If the test statistic falls in the critical region we reject H0
![Page 7: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/7.jpg)
Right-tailed Critical Region:
• α (critical region) occupies the right of the curve• Degree of confidence is in the left of the curve
• Usually used when the H0 < benchmark
![Page 8: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/8.jpg)
Left-tailed critical region
• α (critical region) occupies the left of the curve• Degree of confidence is in the right of the curve
• Usually used when the H0 > benchmark
![Page 9: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/9.jpg)
Two-tailed critical region
• Critical region split between the two ends of the curve
• Degree of confidence is in the center of the curve
• Used when H0 is an equality
![Page 10: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/10.jpg)
Critical Values: Large samples
Degree of confidence
Two tailed Left tailed Right tailed
90 1.645 -1.28 1.28
95 1.96 -1.645 1.645
98 2.33 -2.05 2.05
99 2.575 -2.33 2.33
99.9 3.30 -3.10 3.10
![Page 11: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/11.jpg)
For example
1. H0: μ = 801. DoC = 95%
2. H0: μ <= 1151. DoC = 98%
3. H0: μ >= 701. DoC = 99%
![Page 12: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/12.jpg)
Home Work: Find the Critical Values
1. H0: μ = 401. DoC = 98%
2. H0: μ <= 1151. DoC = 90%
3. H0: μ >= 701. DoC = 96%
4. H1: μ > 981. DoC = 95%
5. H1: μ 0.251. DoC = 95%
6. H0: μ = 4.01. DoC = 99.9%
7. H0: μ ≥ 1.151. DoC = 90%
8. H0: μ ≥ 7.01. DoC = 90%
9. H1: μ = 8.91. DoC = 98%
10. H1: μ ≤ 1.251. DoC = 95%
![Page 13: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/13.jpg)
3. Stating our conclusions
• Strictly speaking, we’ve either reject or fail to reject the null hypothesis.
• We sometimes say “accept” instead of “fail to reject”, but this can be misleading, since we really haven’t proven the null hypothesis; only that there’s not enough evidence to disprove it.
![Page 14: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/14.jpg)
What if we’re wrong?
• Type 1 error: Reject a null hypothesis that is true.– False positive– Probability of doing so is α, the significance level
• Type 2 error: Accept a null hypothesis that is false.– False negative– Probability of doing so is β.
• The consequences of these errors will determine degree of confidence and sample size– If α is fixed, increasing n will decrease β– If n is fixed, then decreasing α will increase β (and vice
versa– Increasing n will decrease both α and β
![Page 15: Proving (or disproving) the Null Hypothesis 1.From the sample and hypothesis benchmark we calculate a test statistic 2.From the hypothesis and degree of.](https://reader036.fdocuments.in/reader036/viewer/2022082400/56649e0d5503460f94af6eaa/html5/thumbnails/15.jpg)
Example
• A vendor claims its batteries can go at least 24 hours before recharging.
• If we pick a 90% degree of confidence…• There’s a 10% probability that we will reject the
claim, even though the actual rate is more that 24 hours
• There’s also a probability we will “accept” the claim even though the rate is less that 24 hours